Variable load sense spring setting for axial piston open circuit pump

ABSTRACT

A variable load sense spring setting for an axial piston open circuit having three subsystems. A backstepping control applied to the first and second subsystems to determine an operator control input and a load sense spool effective open area control input that stabilize the first and second subsystems. The calculating a feasible load sense area from the third subsystem and the control inputs.

CROSS REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application No.61/947,038 filed Mar. 3, 2014.

BACKGROUND OF THE INVENTION

This invention is directed to a hydraulic load sense system, and moreparticularly to a hydraulic load sense system for nonlinearapplications.

Load sense systems are used widely throughout the power hydraulicindustries. One of the main reasons of its popularity is the simplicityof control. Using Load Sense and Pressure Control appropriately, a loadsense system is typically passive in nature which can be seen easilyanalytically. The difference between analytically found solutions andpractical implementations lies in the choice of appropriate parameterssuch as clearance, spool area profile etc. Some of these parametersprescribe overall stability of the pump and in most cases these arefigured out by several trial and error methods.

A complete hydraulic load sense system such as that found in anexcavator is a naturally complex model. Components such as the engine,hydraulic pump, valves, uncertain load dynamics as well as uncertainoperator behavior can instigate instability in various ways in a complexload-sense system. Many have confronted this situation in differentways. Most have ignored some practical aspects such as effect ofload-sense control spool area etc. For a typical load sense system ifthe engine speed is assumed to be constant, the operator input to thevalve is the only external input or effect into the system that drivesthe stability of the whole system. The variation in the load could beconsidered as external disturbance. Even though it is possible toformulate a control algorithm with one input for the whole system, theanalytical solution would be very difficult to obtain because of thecomplexity of the involved nonlinear dynamics. Linearized models aregood enough for predicted load situations, however the controls derivedbased on a linearized model may not be effective as nonlinear controlunder uncertain load conditions. Therefore, there exists a need in theart for a manner in which to address these deficiencies.

Therefore, an objective of the present invention is to prove a hydraulicload sense control that is based upon a nonlinearized model.

Another objective of the present invention is to provide a hydraulicload control actuated by operator command and load-sense spool effectiveopening area.

These and other objectives will be apparent to one of ordinary skill inthe art based upon the following written description, drawings, andclaims.

BRIEF SUMMARY OF THE INVENTION

A hydraulic load sense system for nonlinear applications having a firstsubsystem related to operator command, a second load system related to aload sense spool effective opening area, and third subsystem related toa load sense spool displacement. A first control input to stabilize thefirst subsystem and a second control input to stabilize the secondsubsystem are determined by a computer by applying backstepping controlsto the first and second subsystems.

Using the first and second controls and the third subsystem a feasibleload sense area is calculated by the computer. The computer then usesthe feasible load sense area to control a hydraulic pump.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a side sectional view of a load sense pump;

FIG. 2 is a schematic view of a load sense system;

FIG. 3 is a graph showing a Load Sense spool variable area slot;

FIG. 4 is a graph comparing pump source and load pressure;

FIG. 5 is a graph of a phase plot of pump protection;

FIG. 6 is a graph of system pressure under load oscillation;

FIG. 7 is a graph showing calculated Load Sense spool area;

FIG. 8 is a graph showing operator commandvalve orifice area;

FIG. 9 is a graph showing load flow from source;

FIG. 10 is a graph showing variable margin pressure; and

FIG. 11 is a graph showing cylinder position.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring to FIG. 1, shown is a sectional view of a load sense pump 10.Included in the pump 10 is a servo piston 12 that is slidably receivedwithin a servo piston guide 14. The guide 14 engages a swashplate 16that is operatively connected to a rotor shaft 18. Opposite the servopiston 12, and also engaging the swashplate 16 is a bias piston 20having a bias spring 22.

FIG. 2 shows a schematic view of a hydraulic load sense system 10. Amotor 24 is connected to pump 10. Pump 10 delivers and pressurizes fluidfrom a tank 26 to a control valve 28 at system pressure. As an example,valve 28 is a variable area orifice controlled by operator input. Acompensation circuit 30 is provided that includes a pressure limitingcompensation valve 32 and a load sense compensation valve 34. The systemalso includes a torque control valve 36 such as a servo piston thatadjusts the displacement of the swashplate 16. Connected to the systemis a personal computer 38. Alternatively a load sense control can beused.

The nonlinear model of the hydraulic system is described with thefollowing differential equations. The load sense spool dynamic is givenby

$\begin{matrix}{{\overset{¨}{x}}_{ls} = {\frac{1}{m_{ls}}\left\{ {{c_{ls}{\overset{¨}{x}}_{ls}} - {K_{ls}x_{ls}} + {A_{ls}\left( {P_{s} - P_{ls} - P_{k_{ls}}} \right)}} \right\}}} & (1)\end{matrix}$

where P_(k) _(ls) =load sense spring force per unit area which alsoknown as margin pressure. The Pressure Control spool dynamics is givenby

$\begin{matrix}{{\overset{¨}{x}}_{pc} = {\frac{1}{m_{pc}}\left\{ {{{- C_{pc}}{\overset{.}{x}}_{pc}} - {K_{pc}x_{pc}} + {A_{pc}\left( {P_{s} - P_{k_{pc}}} \right)}} \right\}}} & (2)\end{matrix}$

where P_(k) _(pc) is the Pressure Control spring setting as force perunit area and we will design it later in this paper. The swash plate kitdynamics is given by Equation (3).

I _(a) ä=−C _(a) {dot over (a)}−K _(a) a+T _(mswash) +l _(bs) P _(bs) A_(bs) +l _(bs) K _(bs) x _(bs) −l _(sv) P _(sv) A _(sv)  (3)

T_(mswash) is the swash moments generated by pump piston movements. Weused tabulated test data to interpret this dynamics into simulations.Servo flow in and out of servo is defined as [8]

Q _(stsv) =C _(d) A _(stsv)√{square root over (2/p(P _(s) −P _(sv)))}

Q _(svtt) =C _(d) A _(svtt)√{square root over (2/p(P _(sv) −P ₁))}  (4)

FIG. 3 shows a graph of the variable area slot. For positive x_(ls) theservo flow is defined as flow from source to servo. On the other hand,for negative x_(ls) the flow is defined as servo to tank. The totalhorizontal length of the rectangular area is defined as L_(lsA) and thevertical length is defined as h.

${{Maximum}\mspace{14mu} {opening}\mspace{14mu} {area}} = {\frac{L_{lsA}}{2}h}$

and for a given Load Sense spool displacement the area is defined asx_(ls)h for flow from source to servo and

$\left( {\frac{L_{lsA}}{2} - x_{ls}} \right)$

h as flow from servo to tank. This permits one to calculate a feasibleLoad Sense spool area through Lyapunov stability analysis such that itwould help to calculate the size of choke and gain orifices withrequired. This is the motive of using Load Sense spool area as anexternal input. Now the servo pressure is defined by the followingdifferential equation as

$\begin{matrix}{\mspace{79mu} {{\overset{.}{P}}_{sv} = {\frac{\beta}{V_{sv}}\left( {{c_{d}A_{stsv}\sqrt{\frac{2}{\rho}\left( {P_{s} - P_{sv}} \right)}} - {c_{d}A_{svtt}\sqrt{\frac{2}{\rho}\left( {P_{sv} - P_{1}} \right)}}} \right)}}} & (5) \\{{\overset{.}{P}}_{sv} = {\frac{\beta}{V_{sv}}\left\lbrack {{c_{d}x_{ls}h\sqrt{\frac{2}{\rho}\left( {P_{s} - P_{sv}} \right)}} - {c_{d}{h\left( {\frac{L_{lsA}}{2} - x_{ls}} \right)}\sqrt{\frac{2}{\rho}\left( {P_{sv} - P_{1}} \right)}}} \right\rbrack}} & (6)\end{matrix}$

Note that both the servo and bias piston slide across the swash platefor nonzero swash angle a and therefore the effective length from theswash plate kit center to servo and bias piston contact points aredescribed as

x_(sv)=−l_(sv) tan a

x_(bs)=l_(bs) tan a  (7)

Source flow Q_(s), flow from source to load Q_(stld) and flow fromactuator to tank Q_(ldtt) is defined respectively as

$\begin{matrix}{{Q_{s} = {{\frac{n_{p}A_{p}l_{p}\tan \; a}{\pi}\omega} - Q_{stsv}}}{Q_{stld} = {C_{d}A_{stld}\sqrt{\frac{2}{\rho}\left( {P_{s} - P_{ld}} \right)}}}{Q_{ldtt} = {C_{d}A_{ldtt}\sqrt{\frac{2}{\rho}\left( {P_{ld} - P_{t}} \right)}}}} & (8)\end{matrix}$

Subsequently load pressure is defined as

$\begin{matrix}{{\overset{.}{P}}_{ld} = {\frac{\beta}{V_{ld}}\left( {Q_{stld} - {A_{ld}{\overset{.}{x}}_{ld}}} \right)}} & (9)\end{matrix}$

The hydraulic actuator load dynamics is defined as

{umlaut over (x)}_(ld) =−C _(ld) {dot over (x)} _(ld) −K _(ld) x_(ld)+A_(ld) P _(ld) −m _(ld) g−0.7A _(ld) P _(t)  (10)

Since the cylinder ratio s 7:10 and the rod end of the actuator isdirectly connected to tank, it maintains the tank pressure of P_(t).

$\begin{matrix}{{\overset{.}{P}}_{s} = {\frac{\beta}{V_{P_{s}}}\left( {Q_{s} - Q_{stld} - Q_{leak}} \right)}} & (11)\end{matrix}$

Now defining the states as

x₁=x_(pc), x₂={dot over (x)}_(pc), x₃=P_(s)

x₄=a, x₅={dot over (a)}, x₆=P_(sv)

x₇=x_(ls), X₈={dot over (x)}_(ls),x₉=P_(ld)

x₁₀=x_(ld), x₁₁={dot over (x)}_(ld)  (12)

Rearranging the Equations from (1) to (11) and implying the statevariables one can rewrite the following

$\begin{matrix}{\mspace{79mu} {{\overset{.}{x}}_{1} = x_{2}}} & (13) \\{\mspace{79mu} {{\overset{.}{x}}_{2} = {{\frac{1}{m_{pc}}\left\{ {{{- C_{pc}}x_{2}} - {K_{pc}x_{1}} - {A_{pc}P_{k_{pc}}}} \right\}} + {\frac{A_{pc}}{m_{pc}}x_{3}}}}} & (14) \\{{\overset{.}{x}}_{3} = {{\frac{\beta}{V_{p_{s}}}\left( {{{- C_{d}}A_{stld}\sqrt{\frac{2}{\rho}\left( {x_{3} - x_{9}} \right)}} - Q_{leak}} \right)} + {\frac{\beta}{V_{p_{s}}}\frac{n_{p}A_{p}l_{p}}{\pi}\omega \; {\tan \left( x_{4} \right)}}}} & (15) \\{\mspace{79mu} {{\overset{.}{x}}_{4} = x_{5}}} & (16) \\{{\overset{.}{x}}_{5} = {{\frac{1}{l_{a}}\left\{ {{{- C_{a}}x_{5}} - {K_{a}x_{4}} + T_{mswash} + {l_{bs}A_{bs}x_{3}} + {l_{bs}^{2}K_{bs}{\tan \left( x_{4} \right)}}} \right\}} - {\frac{l_{sv}A_{sv}}{l_{a}}x_{6}}}} & (17) \\{{\overset{.}{x}}_{6} = {\frac{\beta}{V_{sv}}\left\{ {{c_{d}x_{7}h\sqrt{\frac{2}{\rho}\left( {x_{3} - x_{6}} \right)}} - {{c_{d}\left( {\frac{L_{lsA}}{2} - x_{7}} \right)}h\sqrt{\frac{2}{\rho}\left( {x_{6} - P_{t}} \right)}}} \right\}}} & (18) \\{\mspace{79mu} {{\overset{.}{x}}_{7} = x_{8}}} & (19) \\{\mspace{79mu} {{\overset{.}{x}}_{8} = {\frac{1}{m_{ls}}\left\{ {{{- C_{ls}}x_{8}} - {K_{ls}x_{7}} + {A_{ls}\left( {x_{3} - x_{9} - P_{k_{ls}}} \right)}} \right\}}}} & (20) \\{\mspace{79mu} {{\overset{.}{x}}_{9} = {\frac{\beta}{V_{ld}}\left\{ {C_{d}A_{stld}\sqrt{{\frac{2}{\rho}\left( {x_{3} - x_{9}} \right)} - {A_{ld}x_{11}}}} \right\}}}} & (21) \\{\mspace{79mu} {{\overset{.}{x}}_{10} = x_{11}}} & (22) \\{\mspace{79mu} {{\overset{.}{x}}_{11} = {{{- C_{ld}}x_{11}} - {K_{ld}x_{10}} - {A_{ld}x_{9}} + {0.7A_{ld}P_{t}} + {m_{ld}g}}}} & (23)\end{matrix}$

Also considering the virtual state variable as δ₃₉=x₃−x₉ and thus

$\begin{matrix}{{\overset{.}{\delta}}_{39} = {{\frac{\beta}{V_{p_{s}}}\left( {{{- C_{d}}A_{stld}\sqrt{\frac{2}{\rho}\delta_{39}}} - Q_{leak}} \right)} + {\frac{\beta}{V_{p_{s}}}\frac{n_{p}A_{p}l_{p}}{\pi}\omega \; {\tan \left( x_{4} \right)}} - {\frac{\beta}{V_{ld}}\left\{ {C_{d}A_{stld}\sqrt{{\frac{2}{\rho}\delta_{39}} - {A_{ld}x_{11}}}} \right\}}}} & (24)\end{matrix}$

and assuming V_(ps)≈V_(ld)

$\begin{matrix}{{\overset{.}{\delta}}_{39} = {{\frac{\beta}{V_{p_{s}}}\frac{n_{p}A_{p}l_{p}}{\pi}\omega \; {\tan \left( x_{4} \right)}} + {\frac{\beta}{V_{p_{s}}}A_{ld}x_{11}} - {\frac{\beta}{V_{p_{s}}}Q_{leak}} - {\frac{2\; \beta}{V_{p_{s}}}C_{d}A_{stld}\sqrt{\frac{2}{\rho}\delta_{39}}}}} & (25)\end{matrix}$

This is based on the assumption that the volumes V_(ps), V_(ld) are thesame without any loss of generality.

From these equations subsystems are defined. As an example, a firstsubsystem includes equations (21), (22), and (23), a second subsystemincludes equations (15), (16), and (17), and a third subsystem includesequations (13), (14), (19) and (20). The first and the second systemsare actuated by operator commands and load-sense spool effectiveopening. The third subsystem includes Pressure Control and Load Sensespools and is a passive system.

To determine an operator input that would stabilize the first subsystema backstepping control is applied.

In control theory backstepping is used for designing stabilizingcontrols for a special recursive class of nonlinear systems. Startingwith Equation (22) one figures out an appropriate x₁₁ so that Equation(22) will be stabilized. Now, Equation (22) can be written as

{dot over (x)} ₁₀=Θ₁₁(t)  (26)

Θ₁₁(t) is known as the desired control law for subsystem (13) and thechange of variable is z₁₁x₁₁−Θ₁₁(t). Then consider a Lyapunov functioncandidate V₁ as

V ₁=½z ² _(zz)≧0  (27)

Differentiating (27) and as δ₃₉=(x₃−x₉) one has

$\begin{matrix}{\mspace{79mu} {{{\overset{.}{V}}_{1} = {z_{11}{\overset{.}{z}}_{11}}}\mspace{79mu} {{\overset{.}{V}}_{1} = {z_{11}\left( {{\overset{.}{x}}_{11} - {{\overset{.}{\vartheta}}_{11}(t)}} \right)}}{{\overset{.}{V}}_{1} = {z_{11}\left\{ {{{- C_{ld}}x_{11}} - {K_{ld}x_{10}} + {A_{ld}\left( {x_{3} - x_{9}} \right)} + {0.7\; A_{ld}P_{t}} - {A_{ld}x_{3}} + {m_{ld}g} - {{\overset{.}{\vartheta}}_{11}(t)}} \right\}}}{{\overset{.}{V}}_{1} = {z_{11}\left\{ {{{- C_{ld}}x_{11}} - {K_{ld}x_{10}} + {A_{ld}\delta_{39}} + {0.7\; A_{ld}P_{t}} - {A_{ld}x_{3}} + {m_{ld}g} - {{\overset{.}{\vartheta}}_{11}(t)}} \right\}}}}} & (28)\end{matrix}$

Choosing a desired control law δ₃₉ ^(d) as

δ^(d) ₃₉ =A _(ld) ⁻¹(C _(ld) x ₁₁ +K _(ld) x ₁₀−0.7A _(ld) P _(t) +A_(ld) x ₃ −m _(ld) g−γ ₁₁ z _(zz)+{dot over (Θ)}₁₁(t))  (29)

If δ₃₉=δ^(d) ₃₉ then

{dot over (V)}₁=−γ₁₁ z ² ₁₁≦0,γ₁₁>0  (30)

Note that z₁₁ is the error variable for cylinder velocity x₁₁ with thedesired cylinder velocity x^(d) ₁₁; and therefore negative definitenessof (30) will guarantee the tracking. Also note that this processdevelops a tracking problem around cylinder velocity so that for anyoperator command the load velocity could be decided based on guidedtrajectory. This behavior will also solve the problem of cavitation forrunaway loads.

However at this point one cannot guarantee δ₃₉=δ^(d) ₃₉. Thereforeanother error variable is considered as z₃₉=δ₃₉−δ^(d) ₃₉ andδ₃₉=z₃₉+δ^(d) ₃₉ is substituted in the Equation (28); to

{dot over (V)} ₁ =z ₁₁(−C _(ld) x ₁₁ −K _(ld) x ₁₀ +A _(ld) {z ₃₉⇄δ^(d)₃₉}+0.7A _(ld) P _(t) −A _(ld) x ₃ +m _(ld) g−{dot over (Θ)} ₁₁(t))=γ₁₁z ² ₁₁ +A _(ld) z ₁₁ z ₃₉  (31)

Now, for differentiating error variable z₃₉; one gets

$\begin{matrix}\begin{matrix}{{\overset{.}{z}}_{39} = {{\overset{.}{\delta}}_{39} - {\overset{.}{\delta}}_{39}^{d}}} \\{= {{\frac{\beta}{V_{p_{s}}}\frac{n_{p}A_{p}l_{p}}{\pi}\omega \; {\tan \left( x_{4} \right)}} +}} \\{{{\frac{\beta}{V_{p_{s}}}A_{ld}x_{11}} - {\frac{\beta}{V_{p_{s}}}Q_{leak}} - {\frac{2\; \beta}{V_{p_{s}}}C_{d}A_{stld}\sqrt{\frac{2}{\rho}\delta_{39}}} - {\overset{.}{\delta}}_{39}^{d}}}\end{matrix} & (32)\end{matrix}$

At this stage a second Lyapunov function is considered as V₂

V ₂ =V ₁=½z ² ₃₉  (33)

and differentiating (33) one has

$\begin{matrix}\begin{matrix}{{\overset{.}{V}}_{2} = {{\overset{.}{V}}_{1} + {z_{39}{\overset{.}{z}}_{39}}}} \\{= {{{- \gamma_{11}}z_{11}^{2}} + {A_{ld}z_{11}z_{39}} + z_{39}}} \\{\begin{Bmatrix}{{\frac{\beta}{V_{p_{s}}}\frac{n_{p}A_{p}l_{p}}{\pi}\omega \; {\tan \left( x_{4} \right)}} + {\frac{\beta}{V_{p_{s}}}A_{ld}x_{11}} -} \\{{\frac{\beta}{V_{p_{s}}}Q_{leak}} - {\frac{2\; \beta}{V_{p_{s}}}C_{d}A_{stld}\sqrt{\frac{2}{\rho}\delta_{39}}} - {\overset{.}{\delta}}_{39}^{d}}\end{Bmatrix}}\end{matrix} & (34)\end{matrix}$

Note that the proposed input as operator command to the valve hasappeared in (34). For simplicity a simple variable area orifice as thevalve is considered. Therefore operator command is assumed to directlyproportional to the variable area of the orifice A_(stld). Since theterm A_(stid appeared in ()34), an appropriate input is A_(stld) suchthat {dot over (V)}₂≦0. One of the possible choices for A_(stld) couldbe

$\begin{matrix}{A_{stld} = {\left( {\frac{2\; \beta}{V_{p_{s}}}C_{d}\sqrt{\frac{2}{\rho}\delta_{39}}} \right)^{- 1}\left\{ {{A_{ld}\left( {z_{11} + z_{39}} \right)} - {\frac{\beta}{V_{p_{s}}}Q_{leak}} - {\overset{.}{\delta}}_{39}^{d} + {\frac{\beta}{V_{p_{s}}}\frac{n_{p}A_{p}l_{p}}{\pi}\omega \; {\tan \left( x_{4} \right)}} + {\frac{\beta}{V_{p_{s}}}A_{ld}x_{11}}} \right\}}} & (35)\end{matrix}$

so that

{dot over (V)} ₂=−γ₁₁ z ² ₁₁ −A _(ld)γ₃₉ z ² ₃₉≦0  (36)

Equation (36) guarantees the negative semi-definiteness of {dot over(V)}₂, if (35) is feasible and implementable. Note that A_(stld) in (35)is a function of the desired cylinder acceleration. Also, in the firstsubsystem, system pressure, servo pressure and load dynamics areinvolved and a desired valve area is derived in (35) to stabilize thefirst subsystem.

As δ^(d) ₃₉, has been designed, one now can replace P_(k) _(pc) withδ₃₉. Thus the system will have a variable load sense or margin pressuresetting which could be realized by using electronic control of LoadSense spool spring setting. A variable margin also helps to reduce theloss across the valve.

To determine a second input control as effective load-sense spool area,a backstepping technique is applied to the second subsystem. The secondsubsystem involves swashplate 16 and servo 36 dynamics.

Although these subsystems are treated separately for design control,they are interconnected separately and coupled through states. For thesecond subsystem a backstepping control approach is used to find out theappropriate height h(x_(ls)). Starting with (16)

{dot over (x)} ₄ =x ₅ =Θ ₅(t)  (37)

where Θ₅ is the new virtual control law for the second subsystem (16).Therefore the error variable is defined as z₅=x₅Θ₅(t) and similarly aLyapunov function candidate is defined as

V₃=½z² ₅  (38)

Differentiating (38) one has

$\begin{matrix}\begin{matrix}{{\overset{.}{V}}_{3} = {z_{5}{\overset{.}{z}}_{5}}} \\{= {z_{5}\left( {{\overset{.}{x}}_{5} - {{\overset{.}{\vartheta}}_{5}(t)}} \right)}} \\{= {z_{5}\left\lbrack {{\frac{1}{I_{a}}\begin{Bmatrix}{{{- C_{a}}x_{5}} - {K_{a}x_{4}} + T_{mswash} +} \\{{l_{bs}A_{bs}x_{3}} + {l_{bs}^{2}K_{bs}{\tan \left( x_{4} \right)}}}\end{Bmatrix}} - {\frac{l_{sv}A_{sv}}{I_{a}}x_{6}} - {{\overset{.}{\vartheta}}_{5}(t)}} \right\rbrack}}\end{matrix} & (39)\end{matrix}$

Even though the swash moments are nonlinear in nature, this example usedis test branch data for accuracy. Similarly, pump leakage flow Q_(leak)is also computed as tubular data. In (37) and (39), Θ₅ and {dot over(Θ)} ₅ is defined as desired angular velocity and acceleration for theswash plate respectively. Therefore one can choose any desired andguided swash trajectory for the control law derived in (51).

If one defines Θ₆ as

$\begin{matrix}{\vartheta_{6} = {\left( \frac{1_{sv}A_{sv}}{I_{a}} \right)^{- 1}\left\{ {z_{5} - {{\overset{.}{\vartheta}}_{5}(t)} + {\frac{1}{I_{a}}\left( {{{- C_{a}}x_{5}} - {K_{a}x_{4}} + T_{mswash} + {I_{bs}A_{bs}x_{3}} + {1_{bs}^{2}K_{bs}{\tan \left( x_{4} \right)}}} \right)}} \right\}}} & (40)\end{matrix}$

and if x₆=Θ₆ then one can easily find out that

{dot over (V)} ₃=−₆ z ² ₅≦0,y ₆>0  (41)

This does not guarantee x₆=Θ₆ at this point and therefore another errorvariable is used z₆=x₆=Θ₆(t) and substituting x₆=z₆+Θ₆ in (39) one gets

$\begin{matrix}{{\overset{.}{V}}_{3} = {z_{5}\left\lbrack {{\frac{1}{I_{a}}\left\{ {{{- C_{a}}x_{5}} - {K_{a}x_{4}} + T_{mswash} + {l_{bs}A_{bs}x_{3}} + {l_{bs}^{2}K_{bs}{\tan \left( x_{4} \right)}}} \right\}} - {\frac{l_{sv}A_{sv}}{I_{a}}\left( {z_{6} + \vartheta_{6}} \right)} - {{\overset{.}{\vartheta}}_{5}(t)}} \right\rbrack}} & (42) \\{\mspace{79mu} {{\overset{.}{V}}_{3} = {{{- \gamma_{6}}z_{5}^{2}} - {\frac{l_{sv}A_{sv}}{I_{a}}z_{5}z_{6}}}}} & (43)\end{matrix}$

Equation (43) is not negative semi-definite as the sign of the term z₅z₆is unknown. One then proceeds with the next step of backstepping until aproposed input variable shows up in the Lyapunov proof. To continuedifferentiating the variable z₆

$\begin{matrix}{{\overset{.}{z}}_{6} = {{{\overset{.}{x}}_{6} - {{\overset{.}{\vartheta}}_{6}(t)}} = {{\frac{\beta}{V_{sv}}\left\{ {{c_{d}x_{7}h\sqrt{\frac{2}{\rho}\left( {x_{3} - x_{6}} \right)}} - {{c_{d}\left( {\frac{L_{lsA}}{2} - x_{7}} \right)}h\sqrt{\frac{2}{\rho}\left( {x_{6} - P_{t}} \right)}}} \right\}} - {{\overset{.}{\vartheta}}_{6}(t)}}}} & (44) \\{{\overset{.}{z}}_{6} = {{\frac{\beta}{V_{sv}}\left\{ {{c_{d}x_{7}h\sqrt{\frac{2}{\rho}\left( {x_{3} - x_{6}} \right)}} - {{c_{d}\left( {\frac{L_{lsA}}{2} - x_{7}} \right)}h\sqrt{\frac{2}{\rho}\left( {x_{6} - P_{t}} \right)}}} \right\}} - {\left( \frac{l_{sv}A_{sv}}{I_{a}} \right)^{- 1}\frac{}{t}\left\{ {z_{5} - {{\overset{.}{\vartheta}}_{5}(t)} + {\frac{1}{I_{a}}\left( {{{- C_{z}}x_{5}} - {K_{a}x_{4}} + T_{mswash} + {l_{bs}A_{bs}x_{3}} + {l_{bs}^{2}K_{bs}{\tan \left( x_{4} \right)}}} \right)}} \right\}}}} & (45)\end{matrix}$

Rewriting (45) one has

$\begin{matrix}{{\overset{.}{z}}_{6} = {{\frac{\beta}{V_{sv}}\left\{ {{c_{d}x_{7}h\sqrt{\frac{2}{\rho}\left( {x_{3} - x_{6}} \right)}} - {{c_{d}\left( {\frac{L_{lsA}}{2} - x_{7}} \right)}h\sqrt{\frac{2}{\rho}\left( {x_{6} - P_{t}} \right)}}} \right\}} - {\left( \frac{l_{sv}A_{sv}}{I_{a}} \right)^{- 1}\left\{ {z_{5} - {{\overset{.}{\vartheta}}_{5}(t)} + {\frac{1}{I_{a}}\left( {{{- C_{a}}{\overset{.}{x}}_{5}} - {K_{a}{\overset{.}{x}}_{4}} + {\overset{.}{T}}_{mswash} + {l_{bs}A_{bs}{\overset{.}{x}}_{3}} + {l_{bs}^{2}K_{bs}{\sec^{2}\left( x_{4} \right)}}} \right)}} \right\}}}} & (46)\end{matrix}$

Note that Load Sense spool displacement x_(ls) or x₇ is one of thedominant factors in (46). But one can always choose to discretize theservo flow based on x₇ as

$\begin{matrix}{\left. {{\left. {Q_{stsv} = {C_{d}x_{7}h\sqrt{\frac{2}{\rho}\left( {x_{3} - x_{6}} \right)}}} \right\} {\forall{x_{7} > 0}}}{Q_{stsv} = {{C_{d}\left( {\frac{L_{lsA}}{2} - x_{7}} \right)}h\sqrt{\frac{2}{\rho}\left( {x_{3} - P_{t}} \right)}}}} \right\} {\forall{x_{7} \leq 0}}} & (47)\end{matrix}$

Considering another Lyapunov function as

V ₄ =V ₃+½z ² ₆  (48)

and differentiating (48) with respect to time one has

$\begin{matrix}{\mspace{79mu} {{\overset{.}{V}}_{4} = {{{\overset{.}{V}}_{3} + {z_{6}{\overset{.}{z}}_{6}}} = {{{- \gamma_{6}}z_{5}^{2}} - {\frac{l_{sv}A_{sv}}{I_{a}}z_{5}z_{6}} + {z_{6}{\overset{.}{z}}_{6}\mspace{14mu} {or}}}}}} & (49) \\{{\overset{.}{V}}_{4} = {{{- \gamma_{6}}z_{5}^{2}} - {\frac{l_{sv}A_{sv}}{I_{a}}z_{5}z_{6}} + {z_{6}\left\lbrack {{\frac{\beta}{V_{sv}}\left\{ {{C_{d}x_{7}\sqrt{\frac{2}{\rho}\left( {x_{3} - x_{6}} \right)}} - {{C_{d}\left( {\frac{L_{lsA}}{2} - x_{7}} \right)}\sqrt{\frac{2}{\rho}\left( {x_{3} - P_{t}} \right)}}} \right\} h} - {\left( \frac{l_{sv}A_{sv}}{I_{a}} \right)^{- 1}\left\{ {{\overset{.}{z}}_{5} - {{\overset{¨}{\vartheta}}_{5}(t)} + {\frac{1}{I_{a}}\left( {{{- C_{a}}{\overset{.}{x}}_{5}} - {K_{a}{\overset{.}{x}}_{4}} + {\overset{.}{T}}_{mswash} + {l_{bs}A_{bs}{\overset{.}{x}}_{3}} + {l_{bs}^{2}K_{bs}{\sec^{2}\left( x_{4} \right)}}} \right)}} \right\}}} \right\rbrack}}} & (50)\end{matrix}$

The second proposed input h shows up in (50). Therefore one has a chanceto define h such that {dot over (V)}₄<0 and one way to achieve this isas follows

$\begin{matrix}{h = {\left\lbrack {\frac{\beta}{V_{sv}}\left\{ {{C_{d}x_{7}\sqrt{\frac{2}{\rho}\left( {x_{3} - x_{6}} \right)}} - {{C_{d}\left( {\frac{L_{lsA}}{2} - x_{7}} \right)}\sqrt{\frac{2}{\rho}\left( {x_{6} - P_{t}} \right)}}} \right\}} \right\rbrack^{- 1}{\quad\left\lbrack {{\frac{l_{sv}A_{sv}}{I_{a}}z_{5}} - {\gamma_{7}z_{6}} + {\left( \frac{l_{sv}A_{sv}}{I_{a}} \right)^{- 1}\left\{ {{\overset{.}{z}}_{5} - {{\overset{¨}{\vartheta}}_{5}(t)} + {\frac{1}{I_{a}}\left( {{{- C_{a}}{\overset{.}{x}}_{5}} - {K_{a}{\overset{.}{x}}_{4}} + {\overset{.}{T}}_{mswash} + {l_{bs}A_{bs}{\overset{.}{x}}_{3}} + {l_{bs}^{2}K_{bs}{\sec^{2}\left( x_{4} \right)}}} \right)}} \right\}}} \right.}}} & (51)\end{matrix}$

Also using (51) in (50) one has

{dot over (V)} ₄=−γ₆ z ² ₅−γ₇ z ² ₆  (52)

Therefore if h defined in (51) can be realized practically then thesecond subsystem is also stable when swash plate angular velocity andacceleration is guided.

The third subsystem includes Pressure Control and spool dynamics. Whilepassive, the Load Sense spool displacement determines the servo flowwhich controls the pump's swashplate subsystem.

To begin one considers the Load Sense spool dynamics from (19)-(20) andone can rewrite it as

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{x}}_{7} = x_{8}} \\{{\overset{.}{x}}_{8} = {\frac{1}{m_{ls}}\left\{ {{{- C_{ls}}x_{8}} - {K_{ls}x_{7}} + u_{1}} \right\}}}\end{matrix} \right. & (53)\end{matrix}$

where u₁=A_(ls)(x₃−x₉−P_(k) _(ls) ) From [13], the following theoremsare applicable for passive systems.

Theorem 1: The feedback connection of two passive systems is passive.

Theorem 2: Consider a feedback connection of two dynamical systems. Wheninput=0, the origin of the closed-loop system is asymptotically stableif each feedback component is either strictly passive, or outputstrictly passive and zero-state observable. Furthermore, if the storagefunction for each component is radially unbounded, the origin isglobally asymptotically stable.

Theorem 3: consider a feedback connection of a strictly passivedynamical system with a passive memory-less function. When input=0, theorigin of the closed-loop system is uniformly asymptotically stable. Ifthe storage function for the dynamical system is radially unbounded, theorigin will be globally uniformly asymptotically stable.

In view of the Theorems 1-3, one considers a Lyapunov function as

V _(x)=½(K _(ls) x ² ₇ +x ² ₈)  (54)

From this it is easy to prove that while V_(x) is radially unbounded,the origin globally asymptotically stable if u₁=0.

Next consider Pressure Control spool dynamics which is also very similarto Load Sense spool dynamics. Rewriting (2) one has

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{x}}_{1} = x_{2}} \\{{\overset{.}{x}}_{2} = {{\frac{1}{m_{pc}}\left\{ {{{- C_{pc}}x_{2}} - {K_{pc}x_{1}}} \right\}} + u_{2}}}\end{matrix} \right. & (55)\end{matrix}$

with

$u_{2} = {\left( {{\frac{A_{pc}}{m_{pc}}x_{3}} - {\frac{A_{pc}}{m_{pc}}P_{k_{pc}}}} \right).}$

Defining a similar Lyapunov function as

V _(y)=1/2(K _(pc) x ² ₁ +x ² ₂)  (56)

one can prove that the third subsystem is as a whole passive. Forbounded u₁ and u₂ one can also easily show that x₁, x₂, x₇ and x₈ areall bounded. To prove the boundedness of u₁ and u₂ as well as for thewhole system let us consider a Lyapunov function V₅ as

V ₅ =V ₄ +V ₂  (57)

Now differentiating V₅ from (57) we have

{dot over (V)} ₅≦−γ₇ z ² ₅−γ₇ z ² ₆−γ₁₁ z ² ₁₁ −A _(ld)γ₃₉ z ² ₃₉  (58)

if Equations (36) and (51) hold. Therefore the whole load sense systemis bounded and Lyapunov stable.

A simulation was carried out to verify the proposed results. Thesimulation was written as a Matlab script file to solve the stiffdifferential equations given by Equation (1) to (25). Therefore withoutany loss of generality, a simple example was considered which is lessdifficult and complex to implement as a script file.

In the simulation example, the stability of the pump pressure and othervital pump parameters under high frequency load oscillations is shown,and there is an assumption that the load is moving away from the bottomof the cylinder, assumed asx_(id =0.)

FIG. 4 and FIG. 5 show the pump and load pressure characteristics aswell as phase plane plot of the source pressure respectively. It can beseen from the phase plane plot that the system pressure is Lyapunovstable under oscillating load conditions given in FIG. 6.

FIG. 7 and FIG. 8 show the calculated Load Sense spool area and operatorcommand as valve orifice area respectively. It can be noted that, tomaintain practicality of the obtained result, saturations were used forthe inputs and thereby observed whether Lyapunov stability could beachieved under given restrictions. FIG. 9 and FIG. 10 demonstrate theflow and variable margin pressure of the system.

FIG. 11 describes the cylinder position under oscillating loadconditions.

Accordingly, stability analysis for a load sense pump is considered.Backstepping methodology has implemented on a nonlinear pump model toobserve pump stability under oscillating load conditions while computingvalve orifice area and Load Sense spool area as external input. Whilethe former one is a real external input, but the later isn't. Desired isto find out a spool area profile to adjust gain and choke orifice sizeunder oscillating load situation.

What is claimed is:
 1. A method of stabilizing a hydraulic load sensesystem, comprising the steps of: defining a first subsystem related tooperator command, a second subsystem related to a load sense spooleffective opening area, and a third subsystem related to load sensespool displacement; applying with a computer a first backsteppingcontrol to determine a first control input to stabilize the firstsubsystem; applying with a computer a second backstepping control todetermine a second control input to stabilize the second subsystem;calculating a feasible load sense area based upon the first and secondcontrol inputs and the third subsystem; and controlling a hydraulic pumpbased upon the feasible load sense area.
 2. The method of claim 1wherein the first backstepping control includes using a first and asecond error variable.
 3. The method of claim 1 wherein the firstsubsystem includes cylinder acceleration, system pressure, servopressure, and load dynamics.
 4. The method of claim 1 further comprisingthe step of determining a margin pressure setting.
 5. The method ofclaim 1 wherein the first subsystem includes states {dot over (x)}₉,{dot over (x)}₁₀, and {dot over (x)}₁₁.
 6. The method of claim 1 whereinthe second subsystem includes states {dot over (x)}₃, {dot over (x)}₄,and {dot over (x)}₅.
 7. The method of claim 1 wherein the thirdsubsystem includes states {dot over (x)}₁, {dot over (x)}₂, and {dotover (x)}₃.
 8. The method of claim 1 wherein the third subsystem ispassive.